3.4 Properties Of Double Integrals

The following properties hold whenever the integrals exist; the first three are inherited from Riemann sums, and the others follow from them.

1.: \(\iint _D \big (f(x,y) + g(x,y)\big )\,dA = \iint _D f(x,y)\,dA + \iint _D g(x,y)\,dA\).
2.: \(\iint _D c\, f(x,y)\,dA = c \iint _D f(x,y)\,dA\) for every constant \(c\).
3.: If \(f(x,y)\geq g(x,y)\) for all \((x,y)\in D\), then \(\iint _D f(x,y)\,dA \geq \iint _D g(x,y)\,dA\).
4.: If \(D = D_1\cup D_2\), where \(D_1\) and \(D_2\) do not overlap except perhaps on their boundaries, then \[\iint \limits _D f(x,y)\,dA = \iint \limits _{D_1} f(x,y)\,dA + \iint \limits _{D_2} f(x,y)\,dA .\]
5.: Integrating the constant function \(1\) over a region \(D\) gives the area of \(D\): \[\iint \limits _D 1 \,dA = A(D) .\]
6.: If \(m\leq f(x,y) \leq M\) for all \((x,y) \in D\), then \[m\,A(D)\leq \iint \limits _D f(x,y)\,dA \leq M\, A(D) .\]

Property 6 follows from properties 3, 2 and 5.

Example 3.4.1. Use property 6 to estimate \(\iint _D e^{\sin x \cos y}\, dA\), where \(D\) is the disc with centre the origin and radius \(2\).

Solution. Since \(-1 \leq \sin x \leq 1\) and \(-1 \leq \cos y\leq 1\), we have \(-1 \leq \sin x \cos y\leq 1\), and since the exponential is increasing, \[\frac {1}{e} \leq e^{\sin x \cos y} \leq e .\] With \(m = \frac {1}{e}\), \(M = e\) and \(A(D) = \pi (2)^2 = 4\pi \), \[\frac {4\pi }{e}\leq \iint \limits _D e^{\sin x \cos y}\, dA \leq 4\pi e ,\] that is, the integral lies between about \(4.62\) and \(34.2\).

Example 3.4.2. Evaluate \(\iint _D |x - y|\,dA\) over the square \(D = [0,1]\times [0,1]\).

Solution. The integrand changes form across the diagonal \(y = x\), so split \(D\) by property 4 into the triangles \(D_1\) below the diagonal, where \(|x - y| = x - y\), and \(D_2\) above it. By symmetry the two integrals are equal, so \[\iint \limits _D |x - y|\,dA = 2\int _0^1\int _0^x (x - y)\,dy\,dx = 2\int _0^1\frac {x^2}{2}\,dx = \frac 13 .\] This is the mean distance between two numbers chosen at random in \([0,1]\).

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